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Russia's President Dmitry Medvedev ordered an investigation into allegations of voting fraud yesterday, in an attempt to quell public discontent a day after tens of thousands of Russians protested in central Moscow.

"I disagree with the slogans and statements heard at the meetings," Mr Medvedev wrote on his Facebook page yesterday. "Nevertheless, I have given the order that all cases from polling stations that suggest electoral law may have been broken should be investigated."

The Russian protests have been coordinated online using social networks, and it was soon apparent that a lot of people were unconvinced by Mr Medvedev's words. Within two hours, the Facebook post had received over 3,000 mostly negative and often insulting comments. Many expressed cynicism that the investigation would have any real weight.

It is unclear whom Mr Medvedev has tasked with the investigation. The head of Russia's electoral committee, Vladimir Churov, has previously made the unlikely claim that videos uploaded to YouTube that appear to show electoral fraud were filmed in apartments made to look like polling stations.

Prime Minister Vladimir Putin has largely remained silent on the protests that have broken since elections a week ago gave his United Russia party 49 per cent of the vote. His has suggested, however, that they were organised and financed by America, and he even claimed that US Secretary of State Hillary Clinton had "given a signal" to opposition leaders to start protests. Yesterday, Mr Putin's spokesman Dmitry Peskov struck a more conciliatory note: "We respect the point of view of the protesters, we are hearing what is being said, and we will continue to listen to them."

Evgeniya Chirikova, an activist who has become prominent in the opposition movement, published an appeal to Mr Putin yesterday: "Yesterday, you were whistled across the whole country. People whose votes your party stole came out to the streets. Any self-respecting person in this situation should answer the accusations. The people of Russia and the whole world are waiting for your explanation."

She challenged Mr Putin to a debate with Alexei Navalny, an anti-corruption activist who has emerged as the most charismatic of the leaders in the loose anti-regime coalition that has formed in recent days. He did not speak at Saturday's rally, as he was jailed for 15 days after addressing the first in the series of protests last week.

While few Russians, even those who have come out to protest, have much respect for the clique of opposition politicians that have dominated the small, beleaguered liberal movement for the past decade, many have seen something different in Mr Navalny. Not allied to any party, he is charismatic and has a strong Russian nationalist bent which disturbs liberals but appeals greatly to many ordinary Russians.

He has not said that he will try to stand against Mr Putin in the March presidential elections, but if he does there are signs that he may gain the backing of the notoriously fractious opposition as a united anti-Putin candidate. Vladimir Milov, a democratic leader who has fallen out with other opposition figures over strategy in recent weeks, told The Independent at Saturday's protest that he thought Mr Navalny was the best bet for a united candidate. "It's time for Navalny to stop simply flirting with the idea of standing, and declare his candidacy," he said. "We all need to support him."